Baseball’s Arms Race and the Prisoner’s Dilemma

By very popular demand, guest blogger Steve Walters is back. As noted before, Steve teaches economics at Loyola College in Maryland but remains a fan of his hometown Red Sox, Celtics, Patriots, and Bruins.Today he provides insights into the Clemens deal that you will not find anywhere else.

Imagine you’re a police detective and you’ve collared a couple of suspects—call them “Theo” and “Brian”—in a mob hit.They’re in separate interrogation rooms, and now your job is to get enough evidence to put these bad boys away for a long time.

You try to get them to rat each other out, but Theo—the mouthy one with a Boston accent—says “don’t you know about honah among thieves?We ain’t talkin’.”

You point out that you can send each of them upriver for a year (for parole violations) even if they keep quiet.But if one of them turns on the other, you offer to let that slide.They just laugh.Brian, a hard case from the Bronx, says “I can do a year standing on my head.”

Then you get creative.You let each thug start to wonder whether the other is a rat fink. You make each worry about doing life while the other goes into Witness Protection and gets a fresh start.Eventually, both start to crack.

And that’s the essence of what is known as “the prisoner’s dilemma.”Brian thinks “I can shut up and do my year… but what if Theo is ratting me out?I could spend the rest of my life behind bars!But if I rat him out first…”And Theo thinks the same thing. To protect himself against the worst-case scenario (taking the entire fall while the other guy walks) and/or to pursue the best-case scenario (walking while the other guy takes the fall), each punk spills his guts.

Bravo.But what, you ask, does this little discussion of interrogation tactics have to do with more interesting and important topics, like sports?

Well, it just might explain how Rogers Clemens’s agent, Randy Hendricks, got the Yankees to spend such a stupendous amount of money to bring the 44-year-old pitcher out of retirement—yet again.

Let’s first get the cost-benefit analysis out of the way:The Yanks reportedly will pay Clemens roughly $18.5 million for the remainder of the season.(His contract is for $28 million for the year, of which two-thirds will remain once the Rocket is in game shape.)But since the Yankees are already far over the threshold for MLB’s “payroll luxury tax,” they will pay another $7.4 million in payroll taxes as a result of this deal.

So Clemens is costing the Yanks about $25.9 million.Will that investment pay off, financially speaking?No.Freaking.Way.

Make no mistake, considerable extra revenue will flow to the Yankees once they get to start a sure-fire future Hall-of-Famer instead of dreck like Kei Igawa, Darrell Rasner, or Jeff Karstens every fifth day.Just nowhere near $25.9 million extra.

“Market Size, Pay, and Performance: A General Model and Application to Major League Baseball,” by John D. Burger and Stephen J.K. Walters, Journal of Sports Economics, May 2003, v. 4, iss. 2, pp. 108-25.

…and a rosy assumption about how well Clemens will pitch this year, the most optimistic estimate I can make is that he’ll generate no more than $22.5 million in marginal revenue in the Bronx.Probably a lot less.And yes, that includes all likely revenue streams:more viewers on cable TV, enhanced chances of post-season booty, etc.Fantasize about “enhancing the Yankee brand” if you want to, but this deal is a $3.4 million loser for Boss Stein—at best.

But you don’t even have to do this type of math to know that the Yankees overpaid.The other major bidder in the Rocket auction—the Red Sox (Clemens’s hometown Astros apparently stopped talking with him at the end of spring training)—reportedly topped out at a pro-rated $18 million for the year, or no more than $12 million for the rest of the season.How did Hendricks get an extra $6.5 million from the Yanks?He put the “prisoner’s dilemma” to work for his client, expertly playing on the fears of the Yanks’ brass that their arch-rivals would nab Clemens.

In effect, Hendricks held Brian Cashman and Theo Epstein in separate interrogation rooms and made each worry about what the other might do.Here’s a telling episode from Olney’s account:

“The Yankees’ charter touched down in Texas last Monday [4/30] evening, and general manager Brian Cashman sent a text message to Randy Hendricks, the agent for Roger Clemens. ‘I just landed in your wonderful state.’

“Hendricks responded: ‘I’m out of town.’ And a couple of hours later, as Cashman dined with manager Joe Torre, he said, ‘I wonder if he’s at Fenway Park.’

“This was just one more anxious moment in the Yankees’ months-long pursuit of Clemens…”

The two key players in this prisoner’s dilemma-type game had two possible strategies.Bid low for Clemens, and he’d probably stay retired or, perhaps, pitch for the ‘Stros.Either way, that would have been OK for the Sox and Yanks, since then the Rocket wouldn’t have upset the competitive balance between the two A.L. East juggernauts.And both teams are gonna sell out all their remaining games anyway.

From both a baseball and financial perspective, then, bidding low made sense collectively.But not individually.Bidding high for Clemens held out the prospect of taking a competitive advantage in the A.L. East’s “arm’s race”—i.e., it gave the bidder a chance at the best-case scenario, baseball-wise.And it also protected against the worst-case scenario of having the Rocket pitching against you the rest of the year.

Ultimately, the Yankees decided they had to bid high.Very, very high.

Note, however, that this does not mean that Brian Cashman is a fool.Rather, thanks to the way Randy Hendricks set up this auction, Cashman was between a rock and a hard place:bidding high was an unappealing but unavoidable choice.In game theory jargon, it was a “dominant strategy.”Just not a cheap one.

That such situations pop up so often in baseball may be a reason that team owners earn relatively low yields on their investments (in addition to the possibility that they’re utility- rather than profit-oriented, discussed here last week in “Show Me the Wins”).It’s also a reason that savvy agents are well worth their money.A question worth pondering is whether a team that understood game theory could avoid these types of dilemmas—and the red ink that usually results.

15 Responses to "Baseball’s Arms Race and the Prisoner’s Dilemma"

A similar “prisoner dilemna” can be made for Matsuzaka (Yanks, Mets, Sox). Sox managed to also figure out that Daisuke was going to sign for anything reasonable, as going back to Japan was not going to be an option.

Anyway, I presume your playoff generation revenue was based on start-of-season, where the Sox had a bit less than 50% chance of making the playoffs, while the Yanks had more than 50% chance.

As it stands, when Clemens signed, the Yanks were closer to 40%, and the Redsox at 90%. So, Yanks would be able to leverage him more.

Also, the chances of each making the playoffs is dependent on the other, as you have one division winner (and the wild card).

Finally, there’s the Picasso-factor. This is likely Clemens’ last season (yet again), and leveraging the Clemens masterpiece in this manner is also worth something.

On a purely performance standpoint, I have Clemens worth $16MM. With the leveraging capabilities of NY, YES, playoffs, and Picasso, I don’t think it could be that hard to create a model that shows Clemens worth $26MM.

I wonder if this is exacerbated because it is Yanks-Sox or if all teams work this way. I mean, in the prisoner scenario they have no choice but to not know the other’s intentions. But the teams could make the choice to work together on things, or at the very least share some information.

I may be ignorant about big business, but wouldn’t it make financial sense for the teams to band together to keep these agents from having so much power?

For example, if Drew Rosenhaus or Scott Boras weren’t able to leverage their power for giant (sometimes ridiculous) contracts, it would help all the teams save money, wouldn’t it?

On top of it being a situation where a cheater, a team that, after all others agreed to only pay a set amount decided to outbid everyone else, the teams are specifically prohibited from getting together to thwart contract inflation in this way. This is called collusion, when parties get together to fix bids. It’s both against the collective bargaining agreement (and, I believe, at least some degree against the law).

Teams can get around this by publicly making their intents known (e.g. a GM stating on the record that they will only pay a certain total payroll and only X dollars for player Y) and other teams can react to that, but they cannot get together for the sole intent of keeping everyone’s salaries down. It’s happened before (and MLB owners lost lawsuits accordingly) and probably happens to some degree still, but it’s still not allowed under the rules governing baseball.

I get the point of the article, but in drawing the analogy to the prisoner’s dilemma, why are we focusing on what’s better collectively? Sure, each prisoner doing one year is collectively better than one life sentence (assuming, I guess, that the sentenced prisoner lives more than two years), but since when do Epstein and Cashman make decisions based on what decision helps both franchises the most?

The actual best case for each individually, is escaping jail time. This is why the analogy is flawed; in the original prisoner’s dilemma, if both keep their mouth shut, they both escape prison time (here, then, that actually is the best situation both individually and collectively). If one rats, he gets one year rather than life, but this is guarding against the worst-case scenario.

Drawing that to the Clemens scenario isn’t really possible, because we can’t take into account what the subjective value of beating the Red Sox is to someone like Steinbrenner, who has more money than God. And if you believe then that it might be worth it to Steinbrenner even if it hurts the bottom line, then the analogy of the best case scenario being neither team signing him doesn’t fit.

Re: Tango’s “A similar “prisoner dilemna” can be made for Matsuzaka .” YES! Great point. It raises the following question:

Is this really a “repeated game” between the Sox and Yanks, and is Cashman using a “tit-for-tat” strategy, in which he always does what his opponent did previously?

The Sox crushed the other bidders with their posting fee for Dice-K. They bid high, everybody else bid low (relatively speaking). Now the Yanks overbid for the next big FA pitcher. Hmmm.

In some circumstances, such retaliatory “tit-for-tat” bidding can induce cooperation from a rival. If high bidding is generally punished, the idea is that eventually the high bidder will adopt more cooperative low-bidding, which will be reinforced by “tit-for-tat.” It will be interesting to see what happens in the next contest between these two.

The Prisoner’s Dilemma is an interesting and powerful concept, but I’m not sure how it applies here. The idea is that two parties have a shared interest in a particular outcome, but cannot engage in collective action to achieve that outcome if prevented from communicating. But the Sox and Yankees don’t have a shared interest, they are competitors in what is essentially a zero-sum contest: whatever helps one by definition harms the other. Whoever gets Clemens comes out ahead — there is no outcome that leaves both teams better off.

Now yes, both teams would like to pay players less. If they could both agree to limit Clemen’s compensation, and let a coin toss decide who got him, they would save $$. But this would be illegal collusion, as noted in a post above. And still one team would lose out in W-L terms.

I think by the time you finished writing you answered your own question, Guy. Yes, the “shared interest” is the price the teams pay for talent. Cooperative behavior (or collusion) would reduce returns to players and increase returns to teams. It would not affect the aggregate amount of talent available to teams (unless the collusively-set wage rate fell below that in players’ alternative occupations). But teams try to affect the distribution of talent, and steal a competitive advantage, a la the “rat fink” strategy in the prisoner’s dilemma game.

I think the Dice-K situation is a little different. Their the Red Sox did something creative to get their man. Go high on the posting fee and low (ish) on the contract — overall, probably fair value given other FA movements.

This was different. The Sox probably knew that the Yanks would bid more. In fact, as Tango said the Yanks need him WAY more than the Sox do. I’d say the Sox were 90% sure the Yanks would bid high but in case they didn’t it made sense for the Sox to put a fair bid in — which they did.

Steve: Perhaps I’m being dense, but I still don’t see how this qualifies as a prisoner’s dilemma situation. (Or if it does, then every time that two or more employers compete for the services of an employee, but are not permitted to collude, it would qualify.) The essence of the dilemma is that a “win-win” outcome exists, but is prevented by the communication barrier. But for these two teams, there is no win-win situation even if they were permitted to collude. It isn’t in the Yankees’ interest to reduce Clemens’ salary, if it means they may not get him (as it must). Any scenario where the Sox got Clemens would be a Yankees loss, in both baseball and economic terms.

What is true, I think, is that the inability of the Sox and Yanks to communicate creates the potential for Clemens to earn a kind of super-profit. In a public auction for his services, NY would only have to pay $1 more than Boston’s top bid. Here, they appear to have paid about $8M more than necessary. But again, how is that different from nearly every hiring decision made in America? A prospective employee claims to have an offer of X from another employer, and you have to decide whether to believe it or call their bluff. It’s just that the number of $$ here is a lot bigger.

If he’s really not going to bring returns that approach his salary, the win-win would be for both teams to pass on him and let another team (preferably one who would still be unlikely to see the post-season) spend the money.

Jason: that’s a win for the Sox (who already have about a 90% chance of making playoffs), but not the Yankees. Perhaps they are wrong, but they obviously believe that adding Clemens significantly increases their chances of making the postseason.

I think we’re confounding two different commodities here and that’s going to lead to some problems. If the goal is to win, then yes, the Sox do better if neither sign Clemens (provided he can actually pitch effectively). However, the *cost* when people talk about the team *paying too much* isn’t a cost in terms of wins, but in terms of American greenbacks. Since all the talk on the Yankees overpaying revolves around them losing cash-money, that seems to be a more reasonable commodity to compare than playoff probabilities.

I should have said it’s *possibly* win-win if neither team gets him, since I do not know presently if either or both teams are going to make money. Win-win does not imply that both teams profit equally from the matter.

But if we start looking at another commodity, the equation changes. If the only way to win is to make the playoffs, then the dollar figures should be tossed out the door.

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